CTJan27 Online Year 7 - Introduction to Linear Relations

The Coordinate Plane

The Cartesian coordinate system is a grid system used to locate points in a two-dimensional plane. It is also commonly called the coordinate plane.

It is formed by the intersection of two perpendicular number lines:

• x-axis: The horizontal number line.
• y-axis: The vertical number line.

The Origin

The point where the x-axis and y-axis intersect is called the origin. The coordinates of the origin are $(0, 0)$.

Ordered Pairs

Any point on the plane can be located using an ordered pair of numbers, $(x, y)$.

• The first number, $x$, is the x-coordinate or abscissa. It tells you the horizontal distance and direction from the y-axis (how far to move left or right).
• The second number, $y$, is the y-coordinate or ordinate. It tells you the vertical distance and direction from the x-axis (how far to move up or down).

Example: To plot the point $(4, -2)$, you start at the origin, move 4 units to the right (positive x-direction), and then 2 units down (negative y-direction).

The Four Quadrants

The axes divide the plane into four regions called quadrants. They are numbered using Roman numerals (I, II, III, IV) in a counter-clockwise direction starting from the top right.

• Quadrant I: Top right. Both coordinates are positive $(+, +)$.
• Quadrant II: Top left. The x-coordinate is negative, and the y-coordinate is positive $(-, +)$.
• Quadrant III: Bottom left. Both coordinates are negative $(-, -)$.
• Quadrant IV: Bottom right. The x-coordinate is positive, and the y-coordinate is negative $(+, -)$.

Points on the Axes

• If a point has a y-coordinate of 0, like $(x, 0)$, it lies on the x-axis.
• If a point has an x-coordinate of 0, like $(0, y)$, it lies on the y-axis.

What is a Linear Relation?

A linear relation is a relationship between two variables, typically $x$ and $y$, that forms a straight line when graphed. A key feature of a linear relation is its constant rate of change. This means that for every constant increase in the $x$-variable, the $y$-variable also increases or decreases by a constant amount.

Tables of Values

We can represent a relation using a table of values, which pairs $x$-coordinates with their corresponding $y$-coordinates. To determine if the relationship is linear without graphing, we can analyze the values in the table.

What are First Differences?

To check if a relation given in a table is linear, we calculate the first differences. The first differences are the differences between consecutive $y$-values in a table. For this method to work, the differences between the consecutive $x$-values must be constant.

Calculation: First Difference = $y_2 - y_1$, then $y_3 - y_2$, and so on.

The Rule for Linearity

• If the first differences are constant (the same value every time), then the relation is linear.
• If the first differences are not constant, then the relation is non-linear.

Example 1: A Linear Relation

Consider the table below. The $x$-values increase by 1 each time.

Since the first differences are all equal to 3, the relation is linear.

Example 2: A Non-Linear Relation

Consider this table. The $x$-values also increase by 1 each time.

Here, the first differences are 1, 3, and 5. Since these values are not constant, the relation is non-linear.

Introduction to Linear Equations

A linear equation is an equation that represents a straight line on a graph. There are several ways to write the equation of a line. One of the most common and useful formats is the Standard Form.

What is Standard Form?

The standard form of a linear equation is written as:

$Ax + By + C = 0$

Here's what each part means:

• $x$ and $y$ are the variables, representing the coordinates on a Cartesian plane.
• $A$, $B$, and $C$ are integer coefficients (they cannot be fractions or decimals).
• $A$ and $B$ cannot both be zero. At least one of them must be non-zero.
• By convention, the coefficient $A$ is usually a non-negative integer (i.e., $A \ge 0$).

Example: Consider the equation $3x + 4y - 12 = 0$. This is in standard form. Here:

• $A = 3$
• $B = 4$
• $C = -12$

Converting to Standard Form

Often, you'll encounter linear equations in other forms, like the slope-intercept form ($y = mx + b$), and you'll need to convert them to standard form. The goal is to move all terms to one side of the equation, leaving zero on the other side, and then eliminate any fractions.

Example 1: Converting from Slope-Intercept Form

Let's convert the equation $y = 2x + 5$ to standard form.

1. Move all terms to one side: Subtract $y$ from both sides to get all terms on the right. $0 = 2x - y + 5$

2. Rearrange: Rewrite the equation so it matches the $Ax + By + C = 0$ format. $2x - y + 5 = 0$

3. Identify coefficients: Now we can see that $A = 2$, $B = -1$, and $C = 5$.

Example 2: Dealing with Fractions

Let's convert the equation $y = -\frac{1}{3}x + 2$ to standard form.

1. Move all terms to one side: $\frac{1}{3}x + y - 2 = 0$

2. Eliminate the fraction: The coefficients $A$, $B$, and $C$ must be integers. To get rid of the fraction, multiply the entire equation by the denominator, which is 3. $3 (\frac{1}{3}x + y - 2) = 3(0)$ $(3 \cdot \frac{1}{3}x) + (3 \cdot y) - (3 \cdot 2) = 0$

3. Simplify: $1x + 3y - 6 = 0$ or simply $x + 3y - 6 = 0$

4. Identify coefficients: Now, $A = 1$, $B = 3$, and $C = -6$. The equation is in standard form with integer coefficients.